A Tingley’s type problem in n-normed spaces

Abstract

This paper solves a Tingley’s type problem in n-normed spaces and states that for $$n\ge 2$$ , every n-isometry on the unit sphere of an n-normed space is an n-isometry on the whole space except the origin 0. Also, using analytical approach we give a short proof to show that for $$n\ge 3$$ , any mapping which preserves n-norms of values one and zero is, up to pointwise multiplication by $$\pm 1$$ -, a linear n-isometry. This gives a Wigner-type theorem in n-normed spaces which was proven in a recent paper.